Computing Distributions using Random Walks on Graphs Guy

Computing Distributions using Random Walks on Graphs Guy Kindler Dan Romik DIMACS Weizmann Institute of Science

Computing distributions [Knuth, Yao 76] Given a source of random bits, output a sample with given distribution D.

Formalization Example: Compute D={(“ 0”, 1/2), (“ 1”, 1/4), (“ 2”, 1/4)} “H”=right Interpretation: Computing a distribution using a random walk on a binary tree. “T”=left 0 1 “ 1” 2

Infinite trees are sometimes needed D={(“ 0”, 2/3), (“ 1”, 1/3)} o Requirement: Output is reached with probability 1 o Output Tigh can be reached in expectedt!time [Knuth, Yao 76] Ent(D)+O(1) 0 1 0

Some other models o [Romik ’ 99] time. Generate dist. B from dist. A in optimal Generate unbiased coins from biased ones (when bias is unknown). o [von Neumann ’ 51] o [Keane & O’Brien ’ 94] biased ones. o [Peres & Nacu ’ 03] Generate f(p)-biased coins from p- Generate f(p)-biased in “good time”. Generate f(p)-biased coins from pbiased, using a finite graph. o [Mossel & Peres ’ 03]

Finite state generators [Knuth+Yao]: “ 1” “ 0” Output: 1 1 “ 0” 0…

Finite state generators o Interpretation – binary representation: Generating a random variable on [0, 1] using a random walk on a graph A distribution function is computable, if it is the “ 1” “ 0” output distribution of some smooth/analytic f. s. g. “ 0” Question [Knuth+Yao]: which Output: 1 1 0… distributions are computable? o Definition: o “ 1”

History of the problem o [Knuth+Yao ’ 76] Computable analytic density functions must be polynomials with rational coefficients o [Yao ’ 84] The o [this work] roots of such functions must be rational 1. All functions with above properties can be computed 2. Allowing smooth functions does not add computable functions.

We’ll discuss… [Theorem] Let D be a distribution with density function f. If “ 1” o f is a non-negative polynomial o with rational coefficients o and no irrational roots in [0, 1], “ 1” “ 0” then D is computable. “ 0”

Generating some distributions o All order statistics of independent uniform distribution: “ 1” “ 0” Generating max(X, Y): o run two f. s. g’s “in parallel” o output the maximum uniform variables o All distributions with density of the form: f(x)=c xm(1 -x)n [Knuth+Yao ’ 76]

More distributions [Knuth+Yao ’ 76]: uniform on [a, b], for a, b rational. Generating max(X, Y): o run two f. s. g’s “in parallel” o output the maximum All distributions with density of the form: f(x)=c (x-a)m(b-x)n 1[a, b](x)

All distributions what is the set of rational mixtures of such functions ? Question: Q. E. D. ! all polynomials with rational coefficients, and no irrational roots in [0, 1] ! Answer: All distributions with density of the form: f(x)=c (x-a)m(b-x)n 1[a, b](x) Proof: Geometric in o easy: if f 1, . . , fk are nature computable, then so is 2. Non-constructive 1. a 1 f 1+…akfk (for ai rational)

Conclusions o We solved the computability problem in the f. s. g. model, for smooth functions. o We have no good bounds on complexity (size of graph) in this model. Open problems o Solve for other computational models (stack automaton? [Yao 84]) o Solve the general computablity question (no smoothness restriction) o Solve the complexity question

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